JPhrases/Duality

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11B. Duality

h -: f&.g asserts that h is the dual of f under (or with respect to) g

m0 =: A=: +/ % # Arithmetic mean
m1 =: H=: A &. (%"_) Harmonic mean
m2 =: M=: A &. (^&p) Generalized mean
a3 =: N=: (^&) (A&.) L-x norm; 3-norm is 3 N y
m4 =: +/ &. (*:"_) @ +. Magnitude | e.g. m4 3j4
a5 =: each=: &.> Each (f each applies f to each box)
m6 =: ^ .: -&.j. Sine
m7 =: sin -: sinh&.j. Tautology
m8 =: tan -: tanh&.j. "
m9 =: sinh -: sin&.j. "
m10=: cosh -: cos& j. "
m11=: tanh -: tan&.j. "
v12=: <. -: >.&.- Tautology
v13=: >. -: <.&.- Tautology
d14=: * -: +&.^. Tautology
d15=: * -: +&.(10&^.) Tautology
v16=: % -: -&.^. Tautology
d17=: + -: *&.^ Tautology
d18=: + -: *&.(10&^) Tautology
v19=: - -: %&.^ Tautology
v20=: %. -: %.&.|: Tautology
v21=: %. -: %.&.(+@|:) Tautology
m22=: +/\ -: +/\.&.|. Tautology
m23=: +/\. -: +/\ &.|. Tautology
a24=: BW=: 1 : 'u./&.#:' Bitwise adverb
m25=: *. BW Bitwise AND. e.g. m25 _1 100 200
m26=: +. BW Bitwise OR e.g. m26 100 200
m27=: ~: BW Bitwise XOR e.g. m27 100 200
m28=: i.&.(p:^:_1) The primes less than n
m29=: totient=: * -.@%@~.&.q: Euler's totient function
m30=: |.&.;: Reverse the words; e.g. m30 'three score and ten years'
n31=: a=: ' abcdefghijklmnopqrstuvwxyz' Space and alphabet
m32=: encrypt=: (#a)&|@>: &. (a&i.) Julius Caesar's cypher. e.g.
m33=: decrypt=: (#a)&|@<: &. (a&i.) decrypt encrypt x=:'from sea to sea'
m34=: J=: 1&|.&.#: Survivor number in the Josephus problem of order n
d35=: ack=: c1`c1`c2`c3 @. (#.@(,&*)) Ackermann's function
d36=: c1=: >:@]
d37=: c2=: <:@[ ack 1:
d38=: c3=: <:@[ ack [ ack <:@]
m39=: 0&ack -: >:&.(3&+) Tautology
m40=: 1&ack -: 2&+&.(3&+) Tautology
m41=: 2&ack -: 2&*&.(3&+) Tautology
m42=: 3&ack -: 2&^&.(3&+) Tautology
m43=: 4&ack -: ^/@#&2&.(3&+) Tautology
m44=: 5&ack -: 3 : '^/@#&2^:(1+y.)&.(3&+) 1' Tautology